8/13/2019 natphi11okt2012

1/8

Lectures on natural philosophy

Structure of Space-Time

Christian Maes

(Dated: October 11, 2012)

As the world appears to possess spatial and temporal aspects, it is an important

question and even a central topic of natural philosophy to reflect on the nature of

that space-time arena.

I. NATURAL PHILOSOPHY

Natural philosophy is a philosophical reflection on nature, and more in particular it is

the study of foundational aspects of the natural sciences. It is mostly developed in the

study of philosophy of physics, but its ambition today also reaches aspects of biology,

chemistry, computer science etc. The intellectual tradition started in Greece in the Milesian

school founded in the 6th century BC, flourishing in the pre-socratic philosophy and in

the Hellenistic period with Archimedes as culminating figure and father of mathematical

physics. From there, scientific culture was mostly preserved in the Arabic and Muslim

world. A new age will only start with the work of Galileo and Newton, which also marks

the beginning of classical mechanics in the 17th century. Newton still writes about the

mathematical principles of natural philosophy, but in the next centuries, the term itself

(natural philosophy) must make room for the nomenclature of more specialized scientific

and philosophical disciplines.

The Greek miracle refers to the rather rapid birth (6th-4th century BC) and develop-

ment of a scientific culture in ancient Greek, steered by a new ambition of explaining nature

through nature. That intellectual project was mostly speculative and theoretical, but dif-

fered from Babylonian or Egyptian efforts that mostly dealt with data-collection and their

fitting with simple models. The Greek nature-philosophers reached much further. Here we

have philosophers discussing nature devoid of supernatural causes or of mythical origins,

presenting the being and working of nature in terms of mechanisms and causes that refer to

methodologically observable quantities. A second meaning of the word nature is found in

8/13/2019 natphi11okt2012

2/8

8/13/2019 natphi11okt2012

3/8

3

natural motion (to be distinguished from forced motion). That is decided by the

structure of space: an untouched sublunar world would segregate in earth, then water, then

air, then fire. We thus have a spherical rocky earth covered by oceans and air. There is a

geometrically privileged center and the universe is a sphere around it. The fifth nature

(the quintessence) is called the aether, whose natural motion is uniform circular around the

center. That motion is realized by the sphere of fixed stars, turning around with perfect

regularity and defining time itself. Astronomers task for the coming centuries would be to

understand the motion of the sun, the moon and the planets as a combination of circular

motions.

The influence of Aristotles way of thinking and of his world view on Western thought

cannot be overestimated, especially after its incorporation in a Christian theology. The

notion of natural place has in general brought about the idea of natural (immanent)

justice and natural living, i.e., according to a given or imposed structure and order of

the universe.

III. GALILEIS NEW SCIENCE

There is no perfect, unchanging sphere around the center of the universe. The outlook

on how the universe looks like changes drastically in the 15th-16th century. To start with

the solar system, GG adopted Copernicus heliocentric hypothesis, and thought of planet

Earth very much as being just a planet. In fact, his discoveries with the telescope helped

to revolutionize astronomy, to accept the Copernican system.

Galileo Galilei (born in 1564, dying in 1642) was very much inspired by Archimedes and

very much natural philosopher, astronomer, and mathematician. Besides to the sciences

of motion, astronomy, and material science, he also contributed to the development of the

scientific method. There, systematic experiment joined with mathematical (geometric) and

quantitative analysis, rejecting teleological arguments.

The main discovery on motion was to unite motion with rest. Before GG motion was

ontologically different from rest. With GG motion is as natural as rest. Galileo reasoned

8/13/2019 natphi11okt2012

4/8

4

that moving objects eventually stop because of a force called friction. The more systematic

development is however Newtons work. For our discussion, two further studies are impor-

tant, 1) free fall, 2) relativity principle.

ad 1) Galileo discovers that the distance fallen is proportional to the square of the time

it has been falling. He uses balls rolling on inclined planes to show that different weights

fall with the same acceleration. Regardless of the angle at which the planes are oriented,

the final height almost always equals the initial height. Reducing the slope of the opposite

incline, the ball rolls a further distance in order to reach that original height. Taking the

elevation at nearly a 0-degree angle, the ball would roll almost forever, like making an effort

to reach the original height. And if the opposing incline was not even inclined at all (that

is, if it were oriented along the horizontal), then ... an object in motion would continue in

motion. In fact, GG thought it would continue in circular motion.

ad 2) Relativity: GG emphasizes the relativity of motion, and the invariance of natural laws

under uniform straight motion.

IV. NEWTONS PROGRAMME

Isaac Newton (born in 1642 or in 1643 depending on the calendar, dies in 1727) was the

culminating figure of the scientific revolution of the 17th century. His Philosophiae Naturalis

Principia Mathematica of 1687 is one of the most important single works in the history of

modern science.

A. First law

Every body preseveres in its state either of rest or of uniform motion in a straight line,

except insofar as it is complelled to change its state by impressed forces.It means every body, apples and moons. Unification: No fundamental diversity. The

force that maintains the moon in orbit about the earth is precisely the same force that

causes an apple to fall from a tree.

No natural motion is ascribed to a body; in fact every body has an innate tendency to

maintainits state of motion. No need for special points in the universe. Rather, the arena

of motion is absolute space. That space has a structure to give meaning to the notion of

8/13/2019 natphi11okt2012

5/8

5

rest and uniform motion in a straight line.

Space and geometry is Euclidean. Absolute space is 3-dimensional E3, infinite in all

directions with the structure of pencil, straightedge and compass. That is to say, there is a

notion of topology so that we know what we mean by continuous curves. There is a notion

of affinity so that we know what we mean by straight lines and there is a notion of metric

so that we can estimate distances. We also need an absolute time ot measure time, which

goes forward.

Newton believes that this 3-dimensional space exists at every moment of time, and

moreover that identically the same points of space persist through time. For a body to be

resting it means to remain in the same points of absolute space. Same thing for uniform

motion in a straight line.

Newton understands that these persisting parts of absolute motion cannot be perceived

by our senses. What we can observe are the relative positions, or the relative times. Of

course the belief in absolute space is not strange, but Newton will appeal to experiment to

try to prove its existence.

B. Second law

The First law formulates the state of motion of every body when there is no external

force: it is either rest or uniform straight motion in and with respect to absolute space.

That is however insufficient to account for the world where bodies are almost never seen in

uniform motion. In the Second law, Newton launches his programme on forces:

The change in motion is proportional to the impressed motive force, and is made along the

straight line on which the force is impressed.

By the first law, we know what a change of motion means. From the second law we say

that to keep a body say in uniform circular motion, there must be always a force (the

centripetal force). The moon is attracted to the earth by such a force, gravity.

Newton goes on by noticing that the very observation of forces thus means that there is

change of motion, hence absolute space. More precisely he is using the famous bucket

experiment to prove the existence of absolute space. In an otherwise empty space, two

8/13/2019 natphi11okt2012

6/8

6

spheres connected by a cord and rotating around their common center of gravity can be

considered as rotating with respect to absolute space. There is absolute rotation, hence

absolute space and time.

If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted,

then filled with water, and held at rest together with the water; after, by the sudden action

of another force, it is whirled about in the contrary way, and while the cord is untwisting

itself, the vessel continues for some time this motion; the surface of the water will at first

be plain, as before the vessel began to move; but the vessel by gradually communicating its

motion to the water, will make it begin sensibly to revolve, and recede by little and little,

and ascend to the sides of the vessel, forming itself into a concave figure...This ascent of the

water shows its endeavour to recede from the axis of its motion; and the true and absolute

circular motion of the water, which is here directly contrary to the relative, discovers itself,

and may be measured by this endeavour. ... And therefore, this endeavour does not depend

upon any translation of the water in respect to ambient bodies, nor can true circular motion

be defined by such translation. ...; but relative motions...are altogether destitute of any real

effect. ...It is indeed a matter of great difficulty to discover, and effectually to distinguish,

the true motions of particular bodies from the apparent; because the parts of that immovable

space in which these motions are performed, do by no means come under the observations

of our senses. Isaac Newton; Principia, Book 1: Scholium.

The rotating bucket argument demonstrates that true rotational motion cannot be defined

as the relative rotation of the body with respect to the immediately surrounding bodies. In

general, true motion and rest cannot be defined as special instances of motion or rest relative

to other bodies, but instead can be defined only by reference to absolute space. Alternatively,

these experiments provide an operational definition of what is meant by absolute rotation,

and do not pretend to address the question of rotation relative to what?.

C. Absolute space

Absolute, true and mathematical time, of itself, and from its own nature flows equably

without regard to anything external, and by another name is called duration: relative, ap-

parent and common time, is some sensible and external (whether accurate or unequable)

8/13/2019 natphi11okt2012

7/8

7

measure of duration by the means of motion, which is commonly used instead of true time

...

Absolute space, in its own nature, without regard to anything external, remains always

similar and immovable. Relative space is some movable dimension or measure of the

absolute spaces; which our senses determine by its position to bodies: and which is vulgarly

taken for immovable space ... Absolute motion is the translation of a body from one absolute

place into another: and relative motion, the translation from one relative place into another.

We have seen already above how absolute space very naturally enters as the arena of

motion according to Newton. Space has an Euclidean structure, but not much of that is

used for the dynamics itself. Space is infinite, totally homogeneous and symmetric and

many debates have since long been held on how anything can occur in such a universe;

there is too much freedom and no particular place can be taken. Other objections are the

metaphysical nature of space and time. To the latter, we can add the Galilean invariance

principle, which slightly but essentially changes the structure of Newtonian space-time into

Galilean space-time. It is a way to eliminate unobservable behavior.

For Newton, the trajectories of bodies in absolute uniform motion in a straight line

through absolute space are represented by straight lines in a space-time diagram. Putting

a force on an object curves the world-line in the direction of the force. Going to a Galilean

picture, we only make differences between straight and curved trajectories. We no longer

speak about space and time separately, but about events happening in space-time, without

need to identify spatial points throughout time.

D. Determinism

We may regard the present state of the universe as the effect of its past and the cause

of its future. An intellect which at a certain moment would know all forces that set nature

in motion, and all positions of all items of which nature is composed, if this intellect were

also vast enough to submit these data to analysis, it would embrace in a single formula the

movements of the greatest bodies of the universe and those of the tiniest atom; for such an

intellect nothing would be uncertain and the future just like the past would be present before

8/13/2019 natphi11okt2012

8/8

8

its eyes. Pierre Simon Laplace, A Philosophical Essay on Probabilities.

E. Conservation laws

Classical mechanics introduces the concept of energy as conserved over Newtonian tra-

jectories.

F. Gravity

Newton discovers the universal law of gravitation.

I have not as yet been able to discover the reason for these properties of gravity from

phenomena, and I do not feign hypotheses.

That one body may act upon another at a distance through a vacuum without the mediation

of anything else, by and through which their action and force may be conveyed from one

another, is to me so great an absurdity that, I believe, no man who has in philosophic

matters a competent faculty of thinking could ever fall into it.